Discriminative Nonnegative Spectral Clustering with Out-of-Sample Extension

Data clustering is one of the fundamental research problems in data mining and machine learning. Most of the existing clustering methods, for example, normalized cut and (k)-means, have been suffering from the fact that their optimization processes normally lead to an NP-hard problem due to the discretization of the elements in the cluster indicator matrix. A practical way to cope with this problem is to relax this constraint to allow the elements to be continuous values. The eigenvalue decomposition can be applied to generate a continuous solution, which has to be further discretized. However, the continuous solution is probably mixing-signed. This result may cause it deviate severely from the true solution, which should be naturally nonnegative. In this paper, we propose a novel clustering algorithm, i.e., discriminative nonnegative spectral clustering, to explicitly impose an additional nonnegative constraint on the cluster indicator matrix to seek for a more interpretable solution. Moreover, we show an effective regularization term which is able to not only provide more useful discriminative information but also learn a mapping function to predict cluster labels for the out-of-sample test data. Extensive experiments on various data sets illustrate the superiority of our proposal compared to the state-of-the-art clustering algorithms.